Book II · Chapter 15

Chapter 15: Dimension Four from Refinement Rays

Page 71 in the printed volume

Book II Part III: Topology and Global Shape

Classical covering dimension yields zero for τ³ (every profinite space is totally disconnected). This chapter replaces covering dimension by τ-dimension: the minimal number of independent refinement rays whose joint cylinders separate all points. The four ABCD rays are independent (I.P08, Book I) and jointly complete (I.T04, Book I), giving dim_τ = 4. The four dimensions split as 1 + 3: one radial direction (D) and three solenoidal directions (A, B, C). The split is forced by the peel-off asymmetry and the constraint lattice — it is a theorem, not a convention.